Part 6 of my Mapping the Kalam series.
Last time, I showed how Craig & Sinclair use the thought experiment of Hilbert’s Hotel to support the proposition, “an actual infinite cannot exist.” (You’ll have to read the other posts in this series for this one to make sense.)
Today, we look at three objections to Craig & Sinclair’s use of Hilbert’s Hotel and see how the authors respond.
Craig & Sinclair claim:
Hilbert’s Hotel is absurd. But if an actual inﬁnite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual inﬁnite is not metaphysically possible.
Graham Oppy’s response to this is to bite the bullet:
…these allegedly absurd situations are just what one ought to expect if there were… physical inﬁnities. [The fan of actual infinites can] embrace the conclusion of [his] opponent’s reductio ad absurdum argument. [For example:] “They thought they had me, but I outsmarted them. I agreed that it is sometimes just to hang an innocent man.”1
Oppy’s response is to accept the absurdities of Hilbert’s Hotel, much as we accept the absurdities of quantum mechanics. The difference may be that we have tons of strong evidence for the absurdities of quantum mechanics, and perhaps we do not have good evidence for the absurdities of the actual infinite. So, say Craig & Sinclair, Oppy’s acceptance of the absurdities of Hilbert’s Hotel does nothing to show us why we should accept them despite their absurdity.
Sobel observes2 that Hilbert’s Hotel brings two seemingly irrefutable propositions into conflict:
(i) There are not more things in a multitude M than there are in a multitude M′ if there is a one-to-one correspondence of their members.
(ii) There are more things in M than there are in M′ if M′ is a proper submultitude of M.
That is common sense. But as Hilbert’s Hotel shows, (i) and (ii) are in conflict if:
(iii) An inﬁnite multitude exists.
The obvious solution, say Craig & Sinclair, is to reject (iii), which seems much less certain than (i) and (ii). But Sobel’s choice, rather surprisingly, is to retain (iii) and reject (ii). But Sobel offers no argument for rejecting the seemingly irrefutable (ii).
Sobel also says that the absurdities of Hilbert’s Hotel “bring out the physical impossibility of this particular inﬁnity of concurrent real things, not its logical impossibility.” But Craig & Sinclair admit the logical possibility of Hilbert’s Hotel, they simply deny its metaphysical possibility. So, Sobel’s rejection of (ii) is less plausible than Craig & Sinclair’s rejection of (iii), and Sobel’s defense of the logical possibility of Hilbert’s Hotel misses its mark.
In response to Hilbert’s Hotel, Yandell claims that subtraction of infinite quantities does not yield contradictions:
Subtracting the even positive integers from the set of positive integers leaves an inﬁnite set, the odd positive integers. Subtracting all of the positive integers greater than 40 from the set of positive integers leaves a ﬁnite (forty-membered) set. Subtracting all of the positive integers from the set of positive integers leaves one with the null set. But none of these subtractions could possibly lead to any other conclusion than each leads to. This alleged contradictory feature of the inﬁ nite seems not to generate any actual contradictions.3
To which Craig & Sinclair respond:
It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are inﬁnite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an inﬁnity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and inﬁnity as the remainder.
So, neither Oppy nor Sobel nor Yandell can avoid the absurdities of Hilbert’s Hotel, which seem to demonstrate that “an actual infinite cannot exist,” which is in turn part of a syllogism defending the KCA’s first premise, “The universe began to exist.”
But if we can find concrete examples of actual infinities in the universe, this would defeat premise 2.11 – “An actual infinite cannot exist” – despite the strength of the analogy of Hilbert’s Hotel. We turn to this discussion next.